# Modeling opinion of a variable that covers several orders of magnitude

A continuous parameter whose uncertainty extends over several orders of magnitude generally cannot be modelled in the usual manner. For example, an expert may consider that a gram of meat could contain any number of units of bacteria from 1 to 10 000 but just as likely to be around 100 or 1 000. If we were to model this estimate using a Uniform(1,10 000) distribution, for example, we would almost certainly not match the expert's opinion of the values of the cumulative percentiles. The expert would probably place the 25, 50 and 75 percentiles at about 10, 100 and 1 000, where our model places them at 2 500, 5 000 and 7 500 respectively. The reason for such a large discrepancy is that the expert is subconsciously making an estimate in log-space, i.e. s/he is thinking of the log10 values: log101= 0, log1010 = 1, log10100 = 2, etc..To match the expert's approach to estimating, the analyst can also work in log space, so the distribution becomes:

Number of units of bacteria = 10^VoseUniform(0,4).

The figure below compares these two interpretations of the expert opinion by looking at the cumulative distributions and statistics they would produce.

The Uniform(1,10 000) has much larger mean and standard deviation than the 10^Uniform(0,4) distribution and an entirely different shape.

If the expert had said instead that there could be between 1 and 10 000 units of bacteria in a gram of meat, but the most likely number is around 500, we would probably have the greatest success in modeling this variable as:

Number of units of bacteria = 10^VosePERT(0,2.7,4)

where log10500 = 2.7.

If the variable is to be modelled as a 10^x type formula described above, it is judicious to compare the cumulative percentiles at a few sensible points with those the expert would expect. Any radical differences would suggest that the expert is not actually thinking in log-space and the Cumulative distribution could be used instead.